Preheat oven to 350°F (175°C). Grease and line six 8-inch cake pans with parchment paper. If you don't have six pans, bake in batches.

In a medium bowl, whisk together flour, baking powder, and salt. Set aside. In a large mixing bowl, beat butter and sugar on medium-high until fluffy (3-5 minutes). Add eggs one at a time, beating well after each. Stir in vanilla extract. On low speed, add flour mixture in three parts, alternating with milk, starting and ending with flour. Mix until just combined.

Divide batter evenly into six bowls. Add food coloring to each bowl, creating red, orange, yellow, green, blue, and purple batters. Mix until colors are vibrant.

Pour each colored batter into prepared pans. Bake for 15-18 minutes, or until a toothpick inserted into the center comes out clean. Cool cakes in pans for 10 minutes, then transfer to wire racks to cool completely.

Beat cream cheese and butter until smooth. Gradually add powdered sugar, beating on low speed until combined. Add vanilla extract and mix until fluffy.

Place the purple layer on a cake board or plate. Spread a thin layer of frosting. Repeat with blue, green, yellow, orange, and red layers, adding frosting between each. Apply a thin crumb coat over the entire cake and refrigerate for 20 minutes. Apply the final coat of frosting. Decorate with sprinkles if desired.